Surface correlation functions of dead-leaves models

The pore-surface and surface-surface correlation functions are structural characteristics that play an important role in theoretical materials science and in small-angle scattering theory. Symbolic expressions for the surface correlation functions are available only for very few models, and we here derive such expressions for the general class of dead-leaves models. Within these models, a two-phase pore-solid structure is created by sequentially and randomly filling space with porelike or solidlike grains that overlap any preexisting structure, in the same way that dead leaves fall on the ground. The obtained mathematical expressions are valid for any grain shape, in arbitrary dimension. The results are illustrated with monodispersed spherical grains as well as with a dead-leaves realization of a Debye random medium. In the latter case, the size distribution of the grains is designed to produce a structure having an exponential two-point correlation function. Compared to Debye random media obtained by numerical reconstruction, the dead-leaves structure has an almost identical surface-surface correlation function, but a distinctly different pore-surface correlation function with different average mean curvature of the pore-solid surface. Using the concepts introduced in the paper, we submit a general analytic expression for the pore-surface and surface-surface correlation functions of the Boolean model of interpenetrating grains, valid for arbitrary grains. We also derive general expressions for the surface correlation functions of mosaic structures.